On the structure theorem of graded components of $\mathcal{F}$-finite, $\mathcal{F}$-modules over certain polynomial ring

Abstract

Let $K$ be a field of characteristic $p>0$, $A=K[[Y]]$ be a power series ring in one variable and $Q(A)$ be the field of fraction of $A$. Suppose that $R=A[X_1,\dots ,X_n]$ is a standard ${\mathbb{N}}^n$-graded polynomial ring over $A$, i.e., $\mathrm{deg}(A)=\underline{0}\in {\mathbb{N}}^n$ and $\mathrm{deg}(X_j)=e_j\in {\mathbb{N}}^n$. Assume that $M={⨁}_{\underline{u}\in {\mathbb{Z}}^n}{M}_{\underline{u}}$ is a ${\mathbb{Z}}^n$-graded $\mathcal{F}$-finite, $\mathcal{F}$-module over $R$. In this article we prove that, $M_{\underline{u}}\cong E(A/YA{)}^{a(\underline{u})}\oplus Q(A{)}^{b(\underline{u})}\oplus A^{c(\underline{u})}$ for some finite numbers $a(\underline{u}),b(\underline{u}),c(\underline{u})\ge 0$. Let for a subset of $U$ of $\mathcal{S}=\{1,\dots ,n\}$, define a block to be the set $\displaystyle\mathcal{B}(U)=\{\underline{u} \in \mathbb{Z}^n \mid u_i \geq 0 \mbox{ if } i \in U \mbox{ and } u_i \leq -1 \mbox{ if } i \notin U \}$. Note that ${\bigcup }_{U\subseteq \mathcal{S}}\mathcal{B}(U)={\mathbb{Z}}^n$. We prove that the sets $\{a(\underline{u})\mid \underline{u}\in {\mathbb{Z}}^n\}$, $\{b(\underline{u})\mid \underline{u}\in {\mathbb{Z}}^n\}$ and $\{c(\underline{u})\mid \underline{u}\in {\mathbb{Z}}^n\}$ are constant on $\mathcal{B}(U)$ for each subset $U$ of $\{1,\dots ,n\}$. In particular, these results holds for composition of local cohomology modules of the form $H_{I_1}^{i_1}(H_{I_2}^{i_2}(\dots H_{I_r}^{i_r}(R)\dots )$ where $I_1,\dots ,I_r$ are ${\mathbb{N}}^n$-graded ideals of $R$. This provides a positive characteristic analogue of the results proved in \cite{TS-23} by the authors in characteristic zero.

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